Finite Group Databasebeta

← #194177

Subgroups — #194177

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8987C214yesmaximal, complemented
7#8986C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C23×Q16—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#462—24no—
14C2×#462—21no—
15C2×#462—21no—
16C2×#462C418yescomplemented
17C2×#462C2212yes—
18C2×#462C2212yes—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103—21no—
23#2355—22no—
24#2355C2212yes—
25Q8×C4⋊C4C2212yes—
26#2345—22no—
27#2345—21no—
28Q8×C22⋊C4—21no—
29#2295—22no—
30#2295—21no—
31#2285—21no—
32#2285C2211yescharacteristic
33C42×Q8—21no—
34C4×#117—21no—
35C4×#117C2211yescharacteristic
36#1990C2214yes—
37#1988C2214yes—
38#1987—22no—
39#1987—22no—
40#1824—22no—
41#1736—21no—
42#1736—21no—
43#1487—24no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C23×Q8—21no—
49C23×Q8D411yescharacteristic
50C22×Q16—28no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—48no—
54C22×M16—21no—
55C23×C8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yes—
63C2×C4⋊Q8C2312yes—
64C2×C4×Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8C2×C414yes—
70C2×C4×Q8D414yes—
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×#117—21no—
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#462—28no—
85#462—28no—
86#462—24no—
87#462—24no—
88#462C2×C418yes—
89#462C2×C418yes—
90#431—22no—
91#431—22no—
92#423—22no—
93#423—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2×C414yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101C2×#103—28no—
102C2×#103—22no—
103C2×#103—22no—
104C2×C22⋊C8—21no—
105C2×C4×C8C2312yes—
106#391—24no—
107#391—22no—
108#389—22no—
109#389—24no—
110#387—24no—
111#387—24no—
112#387—24no—
113#384—22no—
114#384C2312yes—
115#382—22no—
116#382—21no—
117#380—22no—
118#380—22no—
119#380—21no—
120#380D412yes—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—22no—
136#376—21no—
137C43—22no—
138C43—21no—
139#340—28no—
140#340—24no—
141#340—24no—
142Q8⋊C8—24no—
143Q8⋊C8—24no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—24no—
148C22×Q8—22no—
149C22×Q8—42no—
150C22×Q8—44no—
151C22×Q8—44no—
152C22×Q8—41no—
153C22×Q8—42no—
154C22×Q8C22⋊C414yes—
155C22×Q8C2×D411yescharacteristic
156C22×Q8C2×D411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×Q16—28no—
164C2×Q16—28no—
165C2×Q16—432no—
166C2×Q16—48no—
167C2×Q16—48no—
168C2×M16—42no—
169C2×M16—42no—
170C2×M16—42no—
171C22×C8—24no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—44no—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185C4⋊Q8C22×C414yes—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—44no—
197C4×Q8—44no—
198C4×Q8C22⋊C414yes—
199C4×Q8C22⋊C414yes—
200C4×Q8C22×C412yes—
201C4×Q8C22×C412yes—
202C4×Q8C2×D412yes—
203C4×Q8C2×D412yes—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—42no—
208#117—42no—
209#117Q1614yescomplemented
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C42—24no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—24no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42C22×C412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42C2411yescharacteristic
260C4⋊C8—24no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8C22×C414yes—
267#103—28no—
268#103—28no—
269C22⋊C8—42no—
270C4×C8—22no—
271C4×C8—22no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280C24#12011yescharacteristic
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—28no—
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—42no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—84no—
299C2×Q8—88no—
300C2×Q8C2×C22⋊C412yes—
301C2×Q8C2×C22⋊C412yes—
302C2×Q8AGL(1,ℤ/8)14yes—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—82no—
319C22×C4—84no—
320C22×C4C2×C22⋊C412yes—
321C22×C4C4×D412yes—
322C22×C4C4×D412yes—
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4#12312yes—
329C22×C4C2×Q1612yes—
330C22×C4#13512yes—
331C22×C4#13712yes—
332C22×C4#13712yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336Q16—416no—
337Q16—416no—
338M16—84no—
339C2×C8—24no—
340C2×C8—48no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—84no—
372C4⋊C4C2×C22⋊C412yes—
373C4⋊C4C2×C22⋊C412yes—
374C4⋊C4C2×Q1614yescomplemented
375C22⋊C4—24no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—48no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—44no—
390C42—82no—
391C42—84no—
392C42C2×C22⋊C412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42#12112yes—
396C42#12112yes—
397C42#12312yes—
398C42C23×C411yescharacteristic
399C42C22×D411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C23—22no—
403C23—22no—
404C23#38612yes—
405C23#45112yes—
406C23#45212yes—
407C23C2×#12011yescharacteristic
408Q8—24no—
409Q8—22no—
410Q8—22no—
411Q8—22no—
412Q8—48no—
413Q8—42no—
414Q8—44no—
415Q8—44no—
416Q8—88no—
417Q8—84no—
418Q8#41814yes—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—84no—
447C2×C4—88no—
448C2×C4—84no—
449C2×C4#38612yes—
450C2×C4#38612yes—
451C2×C4#38612yes—
452C2×C4C4×Q1614yes—
453C2×C4#44114yes—
454C2×C4#45112yes—
455C2×C4#45212yes—
456C2×C4#46214yes—
457C2×C4#46414yes—
458C2×C4C22×C22⋊C411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4C2×C4×D411yescharacteristic
461C2×C4C2×#12111yescharacteristic
462C2×C4C2×#12111yescharacteristic
463C2×C4C2×#12311yescharacteristic
464C2×C4C22×Q1611yescharacteristic
465C2×C4C2×#13511yescharacteristic
466C2×C4C2×AGL(1,ℤ/8)12yes—
467C2×C4C2×#13711yescharacteristic
468C2×C4C2×#13711yescharacteristic
469C2×C4#57512yes—
470C8—44no—
471C8—82no—
472C8—82no—
473C22—24no—
474C22—24no—
475C22—22no—
476C22—22no—
477C22#187814yes—
478C22C2×#38611yescharacteristic
479C22C2×#45111yescharacteristic
480C22C2×#45211yescharacteristic
481C22#300612yes—
482C22#302212yes—
483C4—22no—
484C4—24no—
485C4—21no—
486C4—21no—
487C4—22no—
488C4—44no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—82no—
493C4—84no—
494C4—84no—
495C4#187814yes—
496C4C2×#41811yescharacteristic
497C4#289611yescharacteristic
498C4C2×C4×Q1611yescharacteristic
499C4C2×#44111yescharacteristic
500C4C2×#46211yescharacteristic
501C4C2×#46411yescharacteristic
502C4#309412yes—
503C2—24no—
504C2#1880112yes—
505C2#1880312yes—
506C2#2091211yescharacteristic
507C2C2×#187811yescharacteristic
508C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #8987C2semidirect
7. #8986C2semidirect
16. C2×#462C4semidirect
209. #117Q16semidirect
374. C4⋊C4C2×Q16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#4622
Q8×#1172
#218342
#213132
C2×#14872
#89872
#89862
#80452
#78912